Further Results on the Riemann Hypothesis for Angular Lattice Sums

TL;DR

This paper extends the analysis of angular lattice sums involving complex powers and trigonometric functions, providing numerical methods and zero-density results related to the Riemann Hypothesis.

Contribution

It introduces a general expression for evaluating angular lattice sums and investigates their zero distributions, linking them to the Riemann zeta and Catalan beta functions.

Findings

Zero density on the critical line matches that of the Riemann zeta and Catalan beta functions.

Numerical evaluation method for sums to arbitrary order.

Properties of zero trajectories in angular lattice sums.

Abstract

We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing those in a previous paper (McPhedran et al, Porc. Roy. Soc., 2008). We give a general expression which permits numerical evaluation of members of the class of sums to arbitrary order. We use this to illustrate numerically the properties of trajectories along which the real and imaginary parts of the sums are zero, and we show results for the first two of a particular set of angular sums which indicate their density of zeros on the critical line of the complex exponent is the same as that for the product of the Riemann zeta function and the Catalan beta function.